A new mathematical analysis for the dynamics of laminar spray diffusion flames in the vicinity of a vortex flow field is presented. The governing equations for a spray evaporating in an unsteady vortex are studied. New similarity solutions are found for the dynamics of the spray and the flame it supports. Analytical solutions for the spray flames are derived using Shvab-Zeldovich parameters, through which the radial evolution of the flames is found. The results based on the solution reveal the significant effects vorticity and droplet evaporation have on the flame dynamics. An extinction analysis is carried out which maps the influence of the evaporation coefficient and vortex intensity on flame extinguishment. A number of competing factors such as vortex intensity and heat loss due to evaporation were found to be responsible for flame extinction. Despite the model's simplicity, its predictions offer new insights into the driving mechanisms of more complex spray-combustion situations in which droplets are evaporating in vortical environments.